The Earth’s atmosphere: seeing, background, absorption & scatteringObservational Astronomy 2019 Part 8 Prof. S.C. Trager

Seeing

All ground-based observatories suffer from a major problem:

light from distant objects must pass through the Earth’s atmosphere

recall from our discussion of optics that the atmosphere has refractive power

Seeing

The refraction itself isn’t the problem: it’s the variation in the index of refraction that causes problems

Wind, convection, and land masses cause turbulence, mixing the layers of different indices of refraction in non-uniform and continuously varying ways

Seeing

This non-uniform, continuous variation in the index of refraction cause the initially plane-parallel wavefronts to tilt, bend, and corrugate

This causes the twinkling we see when we look at stars with our eyes

Any degradation of images by the atmosphere is called seeing

top of atmosphere

turbulent atmosphere

near ground

SeeingTo understand seeing, we need to examine the index of refraction and its dependence of temperature and pressure (and humidity, too)

Cauchy’s formula, extended by Lorentz to account for humidity, gives the excess (over vacuum) index of refraction of air:

where p is the pressure in mbars, T is the temperature in K, and v is the water vapor pressure in mbars

nair � 1 =77.6⇥ 10�6

T

✓1 +

7.52⇥ 10�3

�2

◆⇣p+ 4810

v

T

⌘

SeeingThe dominant terms are

Seeing — perturbations of the transiting wavefront — is caused by changes in nair differentially across the wavefront due to turbulence

Note that water vapor has little effect on n in the optical (except for fog), but can affect columns of air by causing convection: wet air is lighter than dry air!

nair � 1 = 77.6⇥ 10�6pT�1

SeeingModern observatories are built at very dry sites, so the overall effect of humidity is typically unimportant for seeing (but important for differential refraction)

Ignoring v above, then, for an ideal gas and adiabatic conditions,

Therefore the primary source of index of refraction variations in the atmosphere is thermal variations

We care (mostly) about small-scale variations in T: thermal turbulence

dn

dT/ p

T 2

Thermal turbulenceThermal turbulence in the atmosphere is created on a variety of scales by

convection: air heated by conduction with the warm surface of the Earth becomes buoyant and rises, displacing cooler air

humid (wet) air also rises and displaces dry air

first mountain

ridge

ocean

undisturbed wind flow

turbulence

Thermal turbulence

wind shear: high winds — in particular the jet stream — generate wind shear and eddies at various scales, creating a turbulent interface between other layers in laminar (non-turbulent) flow

first mountain

ridge

ocean

undisturbed wind flow

turbulence

Thermal turbulence

disturbances: large land-form variations — like mountains — can create turbulence

first mountain

ridge

ocean

undisturbed wind flow

turbulence

Thermal turbulenceTurbulence occurs when inertial forces dominate over viscous forces in a fluid

This means that turbulent motions — eddies and vortices — dominate over smooth (laminar) flows

This is true for typical wind speeds and length scales

first mountain

ridge

ocean

undisturbed wind flow

turbulence

Thermal turbulenceTurbulence is created by eddie transfer

kinetic energy is deposited on large scales, which creates eddies

these in turn create smaller eddies, and so on...

...until the shears are so large compared to the eddy scale that the viscosity of air takes over and the kinetic energy is turned into heat

This stops the turbulent cascade and smooths out the temperature fluctuations

Thermal turbulenceKolmogorov showed that the energy spectrum of such a turbulent cascade has a characteristic shape

The inertial range follows a spectrum

where κ is the wavenumber or inverse length

log E

log κ

κ–5/3inertial rangeexternal cutoff

internal cutoffE / �5/3

Thermal turbulenceWhen turbulence occurs in an atmospheric layers with a temperature gradient differing from an adiabatic gradient...

one in which no heat is transferred (i.e., the entropy does not increase)

...it mixes air of different temperatures at the same altitude, producing temperature fluctuations following a Kolmogorov spectrum

log E

log κ

κ–5/3inertial rangeexternal cutoff

internal cutoff

Atmospheric layers contributing to turbulence

There are effectively four layers of the atmosphere that affect the seeing, with gradual transitions of properties between them

Atmospheric layers contributing to turbulence

surface or ground layer: ~few to 10’s of meters — up to few arcminutes seeing!

turbulence generated by wind shear due to friction and topographic effects

site geometry, trees, boulders, etc.

best sites have surface layers <10 m

build telescope above this layer

Atmospheric layers contributing to turbulence

planetary boundary layer: <~1 km above sea level — 10’s of arcseconds seeing

both frictional effects and day-night heating-cooling cycles

creates convection up to inversion layer

most observatories built above this layer

Atmospheric layers contributing to turbulence

atmospheric boundary layer: above planetary boundary layer

not dominated by convection but by abrupt changes in topography

Atmospheric layers contributing to turbulence

free atmosphere: seeing ~0.4″ — best sites mostly see this layer

unaffected by frictional influence of ground

large-scale flows

tropical trade winds (at lower altitudes), jet stream (at higher altitudes)

mechanical turbulence

Seeing

To understand turbulence in the context of seeing, we need to translate the Kolmogorov spectrum into a spatial variation

To do this, we define a structure function: in 1D,

where T(x) and T(x+r) are the temperatures at two points separated by r

DT (r) = h[T (x+ r)� T (x)]2i

SeeingFor a Kolmogorov spectrum,

where CT2 is the temperature structure constant and gives the intensity of thermal turbulence

The variation of the index of refraction (of air) is given by the index of refraction structure constant,

DT (r) = C2T r

2/3

C2n = C2

T [77.6⇥ 10�6(1 + 7.52⇥ 10�3��2)pT�2]2

Seeing

There is a characteristic transverse linear scale over which we can consider the atmospheric variations to “flatten out” and in which plane-parallel wavefronts are transmitted

over these scales, the images are “perfect”

This scale is called the Fried parameter r0

SeeingThe Fried parameter is well-described by integrating Cn2 along the line-of-sight through the atmosphere:

where z is the zenith angle and z′ is the distance along the line of sight

A larger Fried parameter is better! Hence the inverse dependence on Cn2

note also the dependence on cos z, so the Fried parameter gets smaller as one looks through more atmosphere

r0 = [1.67��2(cos z)�1

ZC2

n(z0)dz0]�3/5

Seeing

The Fried parameter is inversely proportional to the size of the PSF transmitted by the atmosphere

The FWHM of the observed PSF can be predicted if we know the variation of Cn2 along the line-of-sight:

✓FWHM(00) = 2.59⇥ 10�5��1/5

(cos z)�1

ZC2

n(z0)dz0

�3/5

Seeing

This is a typical Cn2 profile

Note that FWHM is bigger for larger zenith angles and smaller for longer wavelengths

Isoplanaticity and the Fried model

In 1665, Hooke suggested the existence of “small, moving regions of the atmosphere having different refractive powers that act like lenses” to explain scintillation, the “twinkling” of stars

In 1966, Fried showed that this simple model was a reasonable representation of reality...

Isoplanaticity and the Fried model

Assume that at any given moment, the atmosphere acts like an array of small, contiguous, wedge-shaped refracting cells

These locally tilt the wavefront randomly over the size scale of the cells

each cell imposes its own tilt on the wavefront...

...creating isoplanatic patches in which the wave is fairly smooth and has little curvature

Isoplanaticity and the Fried model

Thus, each isoplanatic patch transmits a quality image of the source

The size of this isoplanatic patch is the Fried parameter r0, roughly the diameter (not the radius!) of the patch

This characterizes the seeing

The more turbulence there is, the smaller r0 is

Isoplanaticity and the Fried model

Typically r0≈10 cm in the optical and varies as λ6/5

Seeing is always better at longer wavelengths!

The angular size of the isoplanatic patch is

where h is the height of the primary turbulence layer above the telescope

✓0 ⇠ 0.6r0/h

Seeing effects

There are primary effects of seeing:

Scintillation (“twinkling”)

Image wander (“agitation”)

Image blurring (“smearing”)

Seeing effectsScintillation (“twinkling”) is the result of a varying amount of energy being received over time

intensity changes

Variations in the shape of the turbulent layer result in moments when it acts like a (net) concave lens, focussing the light, and other moments when it acts like a (net) convex lens, defocussing the light

This is only obvious when the aperture is <~r0, like for the human eye

For larger apertures, these effects average out

Note that planets don’t twinkle, because they’re much bigger than θ0

Because it is an interference effect, scintillation is highly chromatic

Seeing effectsImage wander (“agitation”) is the motion of an image in the focal plane due to changes in the average tilt of the wavefront

Because the wavefront is locally plane-parallel, the image can actually be diffraction-limited if the aperture is <~r0

Then the Airy disk is imaged, but it will wander

Larger apertures average over more isoplanatic patches, so this effect is also only present for small apertures

Seeing effectsImage blurring (smearing) dominates for D>r0

Each isoplanatic patch gives rise to its own Airy disk (FWHM ~ λ/D): these are called speckles

Seen together, they have FWHM ~ λ/r0 and give rise to a shimmering blur: https://www.youtube.com/watch?v=vD9vqoMqg5U

If integrating for long times, they will have this size for the long-exposure PSF

In general, the larger the telescope, the more photons it gathers — but the PSF is limited to λ/r0

need adaptive optics (AO) or speckle interferometry

Remember that FWHM ~ λ–1/5, so seeing is always better in the red

Airy disk

Seeing: summaryThe size of the PSF at the detector is caused by the seeing (for long integrations)

The seeing limits the size of the PSF to FWHM ~ λ/r0 instead of the resolution limit FWHM ~ λ/D

since r0≈10 cm in the optical, if D>10 cm, then for a big telescope the “resolution” is the same as a 10 cm telescope

this is why we are so interested in adaptive optics (AO)!

Backgrounds, absorption, scattering, and extinction

In order to get accurate results, we need to correct for the background and for light removed by the atmosphere through absorption and scattering

Many background sources come from the Earth (its inhabitants and its atmosphere)

But perhaps the most important background source is the sunlight reflected from the moon: moonlight

days from new

U B V R I

0 22 22.7 21.8 20.9 19.9

3 21.5 22.4 21.7 20.8 19.9

7 19.9 21.6 21.4 20.6 19.7

10 18.5 20.7 20.7 20.3 19.5

14 17 19.5 20 19.9 19.2

Sky brightness at CTIO

Surfa

ce b

right

ness

in m

ag/s

q.ar

csec

, fro

m N

OAO

new

slette

r #37

Airglow is emission from molecules and some atoms (like O, Na, H) in the atmosphere

this is the source of “sky (emission) lines” we see in spectra taken from ground-based sites

Night sky at a “light-polluted” site

1992PASP..104...76O

1992PASP..104...76O

1992PASP..104...76O

1992PASP..104...76O

Scattered sunlight during twilight is also a problem at the beginning and end of the night

although very useful for flat fields!

We define three “official” twilights according to the Sun’s position with respect to the horizon:

6 degree twilight: “civil twilight” — when you need to turn your headlights on

12 degree twilight: “nautical twilight” — sailors can still see the horizon but also see enough stars to fix a position with a sextant

18 degree twilight: “astronomical twilight” — fully dark sky

Zodiacal light is sunlight scattered off of Solar System dust, very near the plane of the ecliptic

this is the limiting background on moonless nights at a very dark site (and even in space) away from the Galactic plane

Unresolved galaxy and starlight — like in the plane of the Milky Way — can also cause a significant background

In the near- to mid-IR (>2.4 µm), thermal emission from the Earth and the telescope dominate the background

Clouds will reflect light back from the Earth, Sun, and Moon

Finally, light pollution from poorly-regulated human settlements is a serious — and growing — problem for astronomy

World-wide light pollution

The atmosphere not only adds background but also removes flux, decreasing signal-to-noise (or even removing signal altogether)

Absorption and scattering

Atmospheric absorption in the optical is minimal

at λ<3100 Å, O3 (ozone) cuts off the “optical window”

small but annoying regions at ~6800 Å and ~7500 Å from O2 absorption

at λ>8000 Å, H2O vapor causes absorption, defining “windows” at ~1 µm and beyond

Absorption and scattering

In the near- to mid-IR, H2O and CO2 (along with a little NO2 and CH4) make life difficult, except in certain windows

Absorption and scattering

At longer wavelengths, the far-IR is reachable only from space (or very high-flying balloons or airplanes)

The spectrum opens up again at λ>850 µm—1 mm (depending on water vapor), becoming almost totally transparent at cm–m wavelengths

Absorption and scattering

Wavelengths shorter than 3000 Å are only visible from space (or very high-flying balloons or airplanes)

Absorption and scattering

Absorption and scattering

Scattering arises from two major regimes of particle-photon scattering

Absorption and scatteringMolecular scattering, where the scattering radius a≪λ

Rayleigh scattering is dominant, with a cross-section

where N is the particle density and is a function of temperature and pressure, and therefore altitude, etc.

�R(�) =8⇡3

3

(n2 � 1)2

N2�4/ N�2��4

Absorption and scattering

Rayleigh scattering is the main cause of blue extinction and is responsible for the blue sky!

Absorption and scatteringAerosol scattering, where the scattering radius

This is Mie scattering, which has (roughly)

and so is the main cause of red extinction

a & �/10

� / a2��1

Absorption and scatteringAerosols are highly variable from night to night, as they come from

air pollution

“haze” (dust)

volcanic ash

dust storms

etc.

Absorption and scattering

In the summer at La Palma, wind-blown dust (“calima”) from Saharan dust storms is particularly annoying, causing a significant increase in optical extinction

Annual variation at ORM (La Palma)

Long-term variation at ORM

Airmass

In order to determine extinction corrections, we need to define the airmass of our observations

We quote magnitudes at “the top of the atmosphere” top of telescope

top of atmosphere

Xz

one

airm

ass

Airmass

We define the airmass between the top of the telescope and the top of the atmosphere at the zenith as one airmass or X=1

top of telescope

top of atmosphere

Xz

one

airm

ass

Airmass

Define z as the zenith angle (or zenith distance), where

z=90º–altitude

Then in a plane-parallel approximation to the Earth’s atmosphere,

top of telescope

top of atmosphere

Xz

one

airm

ass

X = 1/ cos z = sec z

AirmassThe plane-parallel approximation is good for z<60º

For larger zenith angles, a spherical-shell description is required:

X = sec z�0.0018167(sec z�1)�0.002875(sec z�1)2�0.0008083(sec z�1)3

X=1

z=60º, X=2

z=90º, X=38

AirmassNote that the cosine rule of spherical trigonometry gives

where ϕ is your latitude, δ is the object’s declination, and h is its hour angle (all in radians or degrees)

X=1

z=60º, X=2

z=90º, X=38cos z = sin� sin � + cos� cos � cosh

At 60º, you look through 2 airmasses

At 71º, you look through 3 airmasses

At 90º, you look through 38 airmasses

Atmospheric extinction and photometry

Although this topic actually belongs with our discussion of photometry, we can now discuss the effect of atmospheric extinction on photometry, the measurement of magnitudes

Atmospheric extinction and photometry

Because atmospheric absorption and scattering is a radiative transfer problem, we can characterize it as an optical depth problem

Atmospheric extinction and photometry

Then we can write that the flux remaining after passing through the atmosphere is

where f0 is the original flux of the object above the atmosphere, X is the airmass, and τ is the optical depth

f = f0e�X/⌧

Atmospheric extinction and photometry

Then, in magnitudes,

Therefore, in magnitudes, the loss of light per airmass is linear, with a proportionality constant

called the extinction coefficient

m = �2.5 log10 f

= �2.5 log10 f0 + k�X

k� = �2.5(log10 e)/⌧

Atmospheric extinction and photometry

We will return to the process of determining kλ later...

In the meantime, for the sky over La Palma, on average, the extinction per unit airmass is given to the right...

Clearly, we want to observe objects in the U band as close to overhead as possible!

filter kλ

U 0.55

B 0.25

V 0.15

R 0.09

I 0.06